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A1 LINEAR PROGRAMMING AND OPTIMAL SOLUTIONS A2

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Solving Linear Programs2

In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the

simplex method,proceeds by moving from one feasible solution to another, at each step improving the value

of the objective function. Moreover, the method terminates after a finite number of such transitions.

Two characteristics of the simplex method have led to its widespread acceptance as a computational tool.

First, the method is robust. It solvesanylinear program; it detects redundant constraints in the problem

formulation; it identifies instances when the objective value is unbounded over the feasible region; and it

solves problems with one or more optimal solutions. The method is also self-initiating. It uses itself either

to generate an appropriate feasible solution, as required, to start the method, or to show that the problem has

no feasible solution. Each of these features will be discussed in this chapter.

Second, the simplex method provides much more than just optimal solutions. As byproducts, it indicates

how the optimal solution varies as a function of the problem data (cost coefficients, constraint coefficients,

and righthand-side data). This information is intimately related to a linear program called thedualto the

given problem, and the simplex method automatically solves this dual problem along with the given problem.

These characteristics of the method are of primary importance for applications, since data rarely is known

with certainty and usually is approximated when formulating a problem. These features will be discussed in

detail in the chapters to follow. Beforepresentingaformaldescriptionofthealgorithm,weconsidersomeexamples. Thoughelementary, procedure.

2.1 SIMPLEX METHOD-A PREVIEW

Optimal Solutions

Consider the following linear program:

Maximizez=0x1+0x2-3x3-x4+20,(Objective 1)

subject to: x

1-3x3+3x4=6,(1)

x

2-8x3+4x4=4,(2)

x j≥0(j=1,2,3,4). Note that as stated the problem has a very special form. It satisfies the following:

1. All decision variables are constrained to be nonnegative.

2. All constraints, except for the nonnegativity of decision variables, are stated as equalities.

38

2.1Simplex Method-A Preview 39

3. The righthand-side coefficients are all nonnegative.

4. One decision variable is isolated in each constraint with a+1 coefficient (x1in constraint (1) andx2in

constraint (2)). The variable isolated in a given constraint does not appear in any other constraint, and

appears with a zero coefficient in the objective function.

A problem with this structure is said to be incanonical form. This formulation might appear to be quite

limited and restrictive; as we will see later, however,anylinear programming problem can be transformed so

that it is in canonical form. Thus, the following discussion is valid for linear programs in general.

Observe that, given any values forx3andx4, the values ofx1andx2are determined uniquely by the equalities. In fact, settingx3=x4=0 immediately gives a feasible solution withx1=6 andx2=4.

Solutions such as these will play a central role in the simplex method and are referred to asbasic feasible

solutions. In general, given a canonical form for any linear program, a basic feasible solution is given by

setting the variable isolated in constraintj, called thejthbasic-variable, equal to the righthand side of the

jth constraint and by setting the remaining variables, callednonbasic, all to zero. Collectively the basic

variables are termed abasis.? In the example above, the basic feasible solutionx1=6,x2=4,x3=0,x4=0, is optimal. For any

other feasible solution,x3andx4must remain nonnegative. Since their coefficients in the objective function

are negative, if eitherx3orx4is positive,zwill be less than 20. Thus the maximum value forzis obtained

whenx3=x4=0.

To summarize this observation, we state the:

Optimality Criterion.Suppose that, in a maximization problem, every nonbasic variable has a non-

positive coefficient in the objective function of a canonical form. Then the basic feasible solution given

by the canonical form maximizes the objective function over the feasible region.

Unbounded Objective Value

Next consider the example just discussed but with a new objective function:

Maximizez=0x1+0x2+3x3-x4+20,(Objective 2)

subject to: x

1-3x3+3x4=6,(1)

x

2-8x3+4x4=4,(2)

x j≥0(j=1,2,3,4).

Sincex3now has a positive coefficient in the objective function, it appears promising to increase the value

ofx3as much as possible. Let us maintainx4=0, increasex3to a valuetto be determined, and updatex1 andx2to preserve feasibility. From constraints (1) and (2), x

1=6+3t,

x

2=4+8t,

z=20+3t.?We have introduced the new termscanonical, basis, andbasic variableat this early point in our discussion becausethese terms have been firmly established as part of linear-programming vernacular.Canonicalis a word used in manycontexts in mathematics, as it is here, to mean ''a special or standard representation of a problem or concept,"" usuallychosen to facilitate study of the problem or concept.Basisandbasicare concepts in linear algebra; our use of theseterms agrees with linear-algebra interpretations of the simplex method that are discussed formally in Appendix A.

40 Solving Linear Programs2.1

No matter how largetbecomes,x1andx2remain nonnegative. In fact, astapproaches+∞,zapproaches +∞. In this case, the objective function is unbounded over the feasible region. The same argument applies to any linear program and provides the: Unboundedness Criterion.Suppose that, in a maximization problem, some nonbasic variable has a

positive coefficient in the objective function of a canonical form. If that variable has negative or zero

coefficients in all constraints, then the objective function is unbounded from above over the feasible

region.

Improving a Nonoptimal Solution

Finally, let us consider one further version of the previous problem:

Maximizez=0x1+0x2-3x3+x4+20,(Objective 3)

subject to: x

1-3x3+3x4=6,(1)

x

2-8x3+4x4=4,(2)

x j≥0(j=1,2,3,4). Now asx4increases,zincreases. Maintainingx3=0, let us increasex4to a valuet, and updatex1andx2 to preserve feasibility. Then, as before, from constraints (1) and (2), x

1=6-3t,

x

2=4-4t,

z=20+t.

Ifx1andx2are to remain nonnegative, we require:

and Therefore, the largest value fortthat maintains a feasible solution ist=1. Whent=1, the new solution becomesx1=3,x2=0,x3=0,x4=1, which has an associated value ofz=21 in the objective function. Note that, in the new solution,x4has a positive value andx2has become zero. Since nonbasic variables

have been given zero values before, it appears thatx4has replacedx2as a basic variable. In fact, it is fairly

simpletomanipulateEqs. (1)and(2)algebraicallytoproduceanewcanonicalform,wherex1andx4become

the basic variables. Ifx4is to become a basic variable, it should appear with coefficient+1 in Eq. (2), and

with zero coefficients in Eq. (1) and in the objective function. To obtain a+1 coefficient in Eq. (2), we

divide that equation by 4, changing the constraints to read: x

1-3x3+3x4=6,(1)

14x2-2x3+x4=1.(2?)

Now, to eliminatex4from the first constraint, we may multiply Eq. (2?) by 3 and subtract it from constraint

(1), giving: x

1-34x2+3x3=3,(1?)

2.1Simplex Method-A Preview 41

14x2-2x3+x4=1.(2?)

Finally, we may rearrange the objective function and write it as: (-z)-3x3+x4= -20 (3) and use the same technique to eliminatex4; that is, multiply (2?) by-1 and add to Eq. (1) giving: (-z)-14x2-x3= -21.

Collecting these equations, the system becomes:

Maximizez=0x1-14x2-x3+0x4+21,

subject to: x

1-34x2+3x3=3,(1?)

14x2-2x3+x4=1,(2?)

x j≥0(j=1,2,3,4). Now the problem is in canonical form withx1andx4as basic variables, andzhas increased from

20 to 21. Consequently, we are in a position to reapply the arguments of this section, beginning with

this improved solution. In this case, the new canonical form satisfies the optimality criterion since all

nonbasic variables have nonpositive coefficients in the objective function, and thus the basic feasible solution

x

1=3,x2=0,x3=0,x4=1, is optimal.

The procedure that we have just described for generating a new basic variable is calledpivoting. It is

the essential computation of the simplex method. In this case, we say that we have just pivoted onx4in the

secondconstraint. Toappreciatethesimplicityofthepivotingprocedureandgainsomeadditionalinsight, let

us see that it corresponds to nothing more than elementary algebraic manipulations to re-express the problem

conveniently. First, let us use constraint (2) to solve forx4in terms ofx2andx3, giving: x

4=14(4-x2+8x3)orx4=1-14x2+2x3.(2?)

Now we will use this relationship to substitute forx4in the objective equation: z=0x1+0x2-3x3+?

1-14x2+2x3?

+20, z=0x1-14x2-x3+0x4+21, and also in constraint (1) x

1-3x3+3?

1-14x2+2x3?

=6, or, equivalently, x

1-34x2+3x3=3.(1?)

by pivoting. We may interpret pivoting the same way, even in more general situations, as merely rearranging

the system by solving for one variable and then substituting for it. We pivot because, for the new basic

variable, we want a+1 coefficient in the constraint where it replaces a basic variable, and 0 coefficients in

all other constraints and in the objective function.

Consequently, after pivoting, the form of the problem has been altered, but the modified equations still

at any given feasible solution.

Indeed, the substitution is merely the familiar variable-elimination technique from high-school algebra,

known more formally as Gauss-Jordan elimination.

42 Solving Linear Programs2.1

In summary, the basic step for generating a canonical form with an improved value for the objective function is described as: Improvement Criterion.Suppose that, in a maximization problem, some nonbasic variable has a

positive coefficient in the objective function of a canonical form. If that variable has a positive coefficient

in some constraint, then a new basic feasible solution may be obtained by pivoting.

Recall that we chose the constraint to pivot in (and consequently the variable to drop from the basis) by

determiningwhich basic variablefirst goes to zero as we increase the nonbasic variablex4. The constraint is

selected by taking the ratio of the righthand-side coefficients to the coefficients ofx4in the constraints, i.e.,

by performing theratio test: min?63,44?

Note, however, that if the coefficient ofx4in the second constraint were-4 instead of+4, the values for

x

1andx2would be given by:

x

1=6-3t,

x

2=4+4t,

so that asx4=tincreases from 0,x2never becomes zero. In this case, we would increasex4tot=63=2.

This observation applies in general for any number of constraints, so that we need never compute ratios for

nonpositive coefficients of the variable that is coming into the basis, and we establish the following criterion:

Ratio and Pivoting Criterion.When improving a given canonical form by introducing variablexsinto thebasis, pivotinaconstraintthatgivestheminimumratioofrighthand-sidecoefficienttocorresponding x scoefficient. Compute these ratios only for constraints that have a positive coefficient forxs.

Observe that the valuetof the variable being introduced into the basis is the minimum ratio. This ratio is

zero if the righthand side is zero in the pivot row. In this instance, a new basis will be obtained by pivoting,

but the values of the decision variables remain unchanged sincet=0.

As a final note, we point out that a linear program may have multiple optimal solutions. Suppose that the

optimality criterion is satisfied and a nonbasic variable has a zero objective-function coefficient in the final

canonical form. Since the value of the objective function remains unchanged for increases in that variable,

we obtain an alternative optimal solution whenever we can increase the variable by pivoting.

Geometrical Interpretation

Thethreesimplexcriteriajustintroducedalgebraicallymaybeinterpretedgeometrically. Inordertorepresent

the problem conveniently, we have plotted the feasible region in Figs. 2.1(a) and 2.1(b) in terms of only the

nonbasic variablesx3andx4. The values ofx3andx4contained in the feasible regions of these figures satisfy

the equality constraints and ensure nonnegativity of the basic and nonbasic variables: x

1=6+3x3-3x4≥0,(1)

x

2=4+8x3-4x4≥0,(2)

x

3≥0,x4≥0.

2.1Simplex Method-A Preview 43

44 Solving Linear Programs2.2

Consider the objective function that we used to illustrate the optimality criterion, z= -3x3-x4+20.(Objective 1)

For any value ofz, sayz=17, the objective function is represented by a straight line in Fig. 2.1(a). As

zincreases to 20, the line corresponding to the objective function moves parallel to itself across the feasible

region. Atz=20, it meets the feasible region only at the pointx3=x4=0; and, forz>20, it no longer touches the feasible region. Consequently,z=20 is optimal. The unboundedness criterion was illustrated with the objective function: z=3x3-x4+20,(Objective 2) which is depicted in Fig.2.1(b). Increasingx3while holdingx4=0 corresponds to moving outward from

the origin (i.e., the pointx3=x4=0) along thex3-axis. As we move along the axis, we never meet either

constraint (1) or (2). Also, as we move along thex3-axis, the value of the objective function is increasing to

The improvement criterion was illustrated with the objective function z= -3x3+x4+20,(Objective 3) which also is shown in Fig. 2.1(b). Starting fromx3=0,x4=0, and increasingx4corresponds to moving from the origin along thex4-axis. In this case, however, we encounter constraint (2) atx4=t=1 and

constraint (3) atx4=t=2. Consequently, to maintain feasibility in accordance with the ratio test, we move

to the intersection of thex4-axis and constraint (2), which is the optimal solution.

2.2 REDUCTION TO CANONICAL FORM

To this point we have been solving linear programs posed in canonical form with (1) nonnegative variables,

constraint. Here we complete this preliminary discussion by showing how to transform any linear program

to this canonical form.

1.Inequality constraints

In Chapter 1, the blast-furnace example contained the two constraints:

40x1+10x2+6x3≥32.5.

The lefthand side in these constraints is the silicon content of the 1000-pound casting being produced. The

constraints specify the quality requirement that the silicon content must be between 32.5 and 55.0 pounds.

To convert these constraints to equality form, introduce two new nonnegative variables (the blast-furnace

example already includes a variable denotedx4) defined as: x

5=55.0-40x1-10x2-6x3,

x

6=40x1+10x2+6x3-32.5.

Variablex5measures the amount that the actual silicon content fallsshortof the maximum content that can

be added to the casting, and is called aslack variable;x6is the amount of silicon inexcessof the minimum

requirement and is called asurplus variable. The constraints become:

40x1+10x2+6x3+x5=55.0,

40x1+10x2+6x3-x6=32.5.

Slack or surplus variables can be used in this way to convert any inequality to equality form.

2.2Reduction to Canonical Form 45

2.Free variables

To see how to treat free variables, or variables unconstrained in sign, consider the basic balance equation of

inventory models: x t+It-1=dt+It.?Production in periodt? ?

Inventory

from period(t-1)? ?

Demand in

periodt? ?

Inventory at

end of periodt? In many applications, we may assume that demand is known and that productionxtmust be nonnegative.

or that there is a shortage of goods and some must be produced later. For instance, ifdt-xt-It-1=3, then

I

t= -3 units must be produced later to satisfy current demand. To formulate models with free variables,

we introduce two nonnegative variablesI+tandI-t,and write I t=I+t-I-t

as a substitute forIteverywhere in the model. The variableI+trepresents positive inventory on hand and

I-trepresents backorders (i.e., unfilled demand). WheneverIt≥0, we setI+t=ItandI-t=0, and when I t<0, we setI+t=0 andI-t= -It. The same technique converts any free variable into the difference betweentwononnegativevariables. Theaboveequation,forexample,isexpressedwithnonnegativevariables as:x t+I+ t-1-I- t-1-I+t+I-t=dt.

Using these transformations, any linear program can be transformed into a linear program with nonnega-

tive variables and equality constraints. Further, the model can be stated with only nonnegative righthand-side

values by multiplying by-1 any constraint with a negative righthand side. Then, to obtain a canonical form,

we must make sure that, in each constraint, one basic variable can be isolated with a+1 coefficient. Some

constraints already will have this form. For example, the slack variablex5introduced previously into the

silicon equation,

40x1+10x2+6x3+x5=55.0,

appears in no other equation in the model. It can function as an intial basic variable for this constraint. Note,

however, that the surplus variablex6in the constraint

40x1+10x2+6x3-x6=32.5

does not serve this purpose, since its coefficient is-1.

3.Artificial variables

There are several ways to isolate basic variables in the constraints where one is not readily apparent. One

particularly simple method is just to add a new variable to any equation that requires one. For instance, the

last constraint can be written as:

40x1+10x2+6x3-x6+x7=32.5,

with nonnegative basic variablex7. This new variable is completely fictitious and is called anartificial

variable. Any solution withx7=0 is feasible for the original problem, but those withx7>0 are not

feasible. Consequently, we should attempt to drive the artificial variable to zero. In a minimization problem,

add the penalty-Mx7to the objective function). ForMsufficiently large,x7will be zero in the final linear

programming solution, so that the solution satisfies the original problem constraint without the artificial

variable. Ifx7>0 in the final tableau, then there is no solution to the original problem where the artificial

variables have been removed; that is, we have shown that the problem is infeasible.

46 Solving Linear Programs2.2

Let us emphasize the distinction between artificial and slack variables. Whereas slack variables have

meaning in the problem formulation, artificial variables have no significance; they are merely a mathematical

convenience useful for initiating the simplex algorithm.

and has been incorporated in some linear programming systems. There are, however, two serious drawbacks

to its use. First, we don"t knowa priorihow largeMmust be for a given problem to ensure that all artificial

variables are driven to zero. Second, using large numbers forMmay lead to numerical difficulties on a

computer. Hence, other methods are used more commonly in practice.

An alternative to the bigMmethod that is often used for initiating linear programs is called thephase

I-phaseIIprocedureandworksintwostages. PhaseIdeterminesacanonicalformfortheproblembysolving

a linear program related to the original problem formulation. The second phase starts with this canonical

form to solve the original problem. To illustrate the technique, consider the linear program:

Maximizez= -3x1+3x2+2x3-2x4-x5+4x6,

subject to: x

1-x2+x3-x4-4x5+2x6-x7+x9=4,

-3x1+3x2+x3-x4-2x5+x8=6, -x3+x4+x6+x10=1, x

1-x2+x3-x4-x5+x11????=0,

x j≥0(j=1,2,...,11).Artificial variables added

Assume thatx8is a slack variable, and that the problem has been augmented by the introduction of artificial

variablesx9,x10, andx11in the first, third and fourth constraints, so thatx8,x9,x10, andx11form a basis.

The following elementary, yet important, fact will be useful: solutiontotheoriginalproblem. Conversely,everyfeasiblesolutiontotheoriginalproblemprovidesafeasible solution to the augmented system by setting all artificial variables to zero.

Next, observe that since the artificial variablesx9,x10, andx11are all nonnegative, they are all zero only

whentheirsumx9+x10+x11iszero. Forthebasicfeasiblesolutionjustderived,thissumis5. Consequently,

the artificial variables can be eliminated by ignoring the original objective function for the time being and

minimizingx9+x10+x11(i.e., minimizing the sum of all artificial variables). Since the artificial variables

are all nonnegative, minimizing their sum means driving their sum towards zero. If the minimum sum is 0,

then the artificial variables are all zero and a feasible, but not necessarily optimal, solution to the original

problem has been obtained. If the minimum is greater than zero, then every solution to the augmented system

hasx9+x10+x11>0, so thatsomeartificial variable is still positive. In this case, the original problem has

no feasible solution.

Moreover, adding the artificial variables has isolated one basic variable in each constraint. To complete the

function. Since we have presented the simplex method in terms of maximizing an objective function, for the

phase I linear program we will maximizewdefined to beminusthe sum of the artificial variables, rather than

minimizing their sum directly. The canonical form for the phase I linear program is then determined simply

by adding the artificial variables to thewequation. That is, we add the first, third, and fourth constraints in

the previous problem formulation to: (-w)-x9-x10-x11=0,

2.3 Simplex Method-A Full Example 47

and express this equation as: The artificial variables now have zero coefficients in the phase I objective. Note that the initial coefficients for the nonartificial variablexjin thewequation is the sum of the coefficients ofxjfrom the equations with an artificial variable (see Fig. 2.2).

Ifw=0 is the solution to the phase I problem, then all artificial variables are zero. If, in addition, every

artificial variable is nonbasic in this optimal solution, then basic variables have been determined from the

original variables, so that a canonical form has been constructed to initiate the original optimization problem.

(Some artificial variables may be basic at value zero. This case will be treated in Section 2.5.) Observe that

the unboundedness condition is unnecessary. Since the artificial variables are nonnegative,wis bounded

apply. To recap, artificial variables are added to place the linear program in canonical form. Maximizingw either i) gives maxw <0. The original problem is infeasible and the optimization terminates; or ii) gives maxw=0. Then a canonical form has been determined to initiate the original problem. Apply

the optimality, unboundedness, and improvement criteria to the original objective functionz, starting

with this canonical form.

In order to reduce a general linear-programming problem to canonical form, it is convenient to perform

the necessary transformations according to the following sequence:

1. Replaceeachdecisionvariableunconstrainedinsignbyadifferencebetweentwononnegativevariables.

This replacement applies to all equations including the objective function.

2. Change inequalities to equalities by the introduction of slack and surplus variables. For≥inequalities,

let the nonnegativesurplus variablerepresent the amount by which the lefthand side exceeds the righthand side exceeds the lefthand side.

3. Multiply equations with a negative righthand side coefficient by-1.

4. Add a (nonnegative) artificial variable to any equation that does not have an isolated variable readily

apparent, and construct the phase I objective function.

To illustrate the orderly application of these rules we provide, in Fig. 2.2, a full example of reduction to

canonical form. The succeeding sets of equations in this table represent the stages of problem transformation

as we apply each of the steps indicated above. We should emphasize that at each stage the form of the given

problem is exactly equivalent to the original problem.

2.3 SIMPLEX METHOD-A FULL EXAMPLE

Thesimplexmethodforsolvinglinearprogramsisbutoneofanumberofmethods, oralgorithms, forsolving optimizationproblems. Byanalgorithm,wemeanasystematicprocedure,usuallyiterative,forsolvingaclass

of problems. The simplex method, for example, is an algorithm for solving the class of linear-programming

problems. Any finite optimization algorithm should terminate in one, and only one, of the following possible

situations:

1. by demonstrating that there is no feasible solution;2. by determining an optimal solution; or3. by demonstrating that the objective function is unbounded over the feasible region.

48 Solving Linear Programs2.3

2.3 Simplex Method-A Full Example 49

50 Solving Linear Programs2.3

We will say that an algorithm solves a problem if it always satisfies one of these three conditions. As we shall

see, a major feature of the simplex method is that it solves any linear-programming problem.

Most of the algorithms that we are going to consider are iterative, in the sense that they move from one

decision pointx1,x2,...,xnto another. For these algorithms, we need: i) a starting point to initiate the procedure; ii) a termination criterion to indicate when a solution has been obtained; and iii) an improvement mechanism for moving from a point that is not a solution to a better point. Every algorithm that we develop should be analyzed with respect to these three requirements.

In the previous section, we discussed most of these criteria for a sample linear-programming problem.

Now we must extend that discussion to give a formal and general version of the simplex algorithm. Before

doing so, let us first use the improvement criterion of the previous section iteratively to solve a complete

problem. To avoid needless complications at this point, we select a problem that does not require artificial

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